A wire of length 20.0 cm lies along the x -axis with the center of the wire at the origin. The

step 1 Hi, in the given problem, there is a straight conductor kept along the x -axis and having a leng

A wire of length 20.0 cm lies along the x -axis with the...

A wire of length $20.0 \mathrm{~cm}$ lies along the $x$ -axis with the center of the wire at the origin. The wire carries current $I=8.00 \mathrm{~A}$ in the $-x$ -direction. (a) What is the magnitude $B$ of the magnetic field of the wire at the point $y=5.00 \mathrm{~cm}$ on the $y$ -axis? (b) What is the percent difference between the answer in (a) and the value you obtain if you assume the wire is infinitely long and use Eq. ( 28.9 ) to calculate $B$ ?

Key Concepts

Biot-Savart Law

This fundamental law in magnetism explains how current elements produce magnetic fields. It provides the basis for calculating the magnetic field at any point in space due to a small segment of current-carrying conductor by integrating contributions over the contour of the current. This concept is central to understanding how the geometry and finite extent of a current-carrying wire affect the resulting magnetic field.

Magnetic Field Due to a Finite Wire

When dealing with a wire of limited length, the magnetic field calculation requires integrating the Biot-Savart law over the finite segment, which leads to a formula involving sine functions of the angles that the endpoints of the wire subtend at the observation point. This concept highlights how the finite size of the conductor modifies the field compared to idealized cases and is crucial for accurately calculating magnetic fields in real-world scenarios.

Magnetic Field Due to an Infinite Wire

This idealization assumes that a wire extends infinitely in both directions, which simplifies the magnetic field calculation to a well-known formula: B = (??I) / (2?r). This approximation is often used because it simplifies analysis and provides insight into the behavior of magnetic fields at points relatively close to the wire, though it becomes less accurate when the actual wire length is finite.

Percentage Difference

This concept involves quantifying how much two values differ relative to each other, which in this context is used to compare the magnetic field predicted by the finite-wire formula against the infinite-wire approximation. The percentage difference helps assess the validity of the approximation and emphasizes the impact of the finite geometry on the magnetic field.

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